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Mirrors > Home > MPE Home > Th. List > gcdmultiplezOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of gcdmultiplez 15883 as of 12-Jan-2024. Extend gcdmultiple 15884 so 𝑁 can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
gcdmultiplezOLD | ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (𝑀 · 𝑁)) = 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7164 | . . . 4 ⊢ (𝑁 = 0 → (𝑀 · 𝑁) = (𝑀 · 0)) | |
2 | 1 | oveq2d 7172 | . . 3 ⊢ (𝑁 = 0 → (𝑀 gcd (𝑀 · 𝑁)) = (𝑀 gcd (𝑀 · 0))) |
3 | 2 | eqeq1d 2823 | . 2 ⊢ (𝑁 = 0 → ((𝑀 gcd (𝑀 · 𝑁)) = 𝑀 ↔ (𝑀 gcd (𝑀 · 0)) = 𝑀)) |
4 | nncn 11646 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ) | |
5 | zcn 11987 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
6 | absmul 14654 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁))) | |
7 | 4, 5, 6 | syl2an 597 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁))) |
8 | nnre 11645 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
9 | nnnn0 11905 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0) | |
10 | 9 | nn0ge0d 11959 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 0 ≤ 𝑀) |
11 | 8, 10 | absidd 14782 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (abs‘𝑀) = 𝑀) |
12 | 11 | oveq1d 7171 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → ((abs‘𝑀) · (abs‘𝑁)) = (𝑀 · (abs‘𝑁))) |
13 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) · (abs‘𝑁)) = (𝑀 · (abs‘𝑁))) |
14 | 7, 13 | eqtrd 2856 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 𝑁)) = (𝑀 · (abs‘𝑁))) |
15 | 14 | oveq2d 7172 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (abs‘(𝑀 · 𝑁))) = (𝑀 gcd (𝑀 · (abs‘𝑁)))) |
16 | 15 | adantr 483 | . . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 gcd (abs‘(𝑀 · 𝑁))) = (𝑀 gcd (𝑀 · (abs‘𝑁)))) |
17 | simpll 765 | . . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → 𝑀 ∈ ℕ) | |
18 | 17 | nnzd 12087 | . . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → 𝑀 ∈ ℤ) |
19 | nnz 12005 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
20 | zmulcl 12032 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
21 | 19, 20 | sylan 582 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) |
22 | 21 | adantr 483 | . . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 · 𝑁) ∈ ℤ) |
23 | gcdabs2 15879 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (𝑀 gcd (abs‘(𝑀 · 𝑁))) = (𝑀 gcd (𝑀 · 𝑁))) | |
24 | 18, 22, 23 | syl2anc 586 | . . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 gcd (abs‘(𝑀 · 𝑁))) = (𝑀 gcd (𝑀 · 𝑁))) |
25 | nnabscl 14685 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ) | |
26 | gcdmultiple 15884 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ) → (𝑀 gcd (𝑀 · (abs‘𝑁))) = 𝑀) | |
27 | 25, 26 | sylan2 594 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑀 gcd (𝑀 · (abs‘𝑁))) = 𝑀) |
28 | 27 | anassrs 470 | . . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 gcd (𝑀 · (abs‘𝑁))) = 𝑀) |
29 | 16, 24, 28 | 3eqtr3d 2864 | . 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 gcd (𝑀 · 𝑁)) = 𝑀) |
30 | mul01 10819 | . . . . . 6 ⊢ (𝑀 ∈ ℂ → (𝑀 · 0) = 0) | |
31 | 30 | oveq2d 7172 | . . . . 5 ⊢ (𝑀 ∈ ℂ → (𝑀 gcd (𝑀 · 0)) = (𝑀 gcd 0)) |
32 | 4, 31 | syl 17 | . . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 gcd (𝑀 · 0)) = (𝑀 gcd 0)) |
33 | 32 | adantr 483 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (𝑀 · 0)) = (𝑀 gcd 0)) |
34 | nn0gcdid0 15869 | . . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑀 gcd 0) = 𝑀) | |
35 | 9, 34 | syl 17 | . . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 gcd 0) = 𝑀) |
36 | 35 | adantr 483 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 0) = 𝑀) |
37 | 33, 36 | eqtrd 2856 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (𝑀 · 0)) = 𝑀) |
38 | 3, 29, 37 | pm2.61ne 3102 | 1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (𝑀 · 𝑁)) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 · cmul 10542 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 abscabs 14593 gcd cgcd 15843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 |
This theorem is referenced by: (None) |
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